Underscreened Kondo effect in quantum dots coupled to ferromagnetic leads

We analyze the equilibrium transport properties of underscreened Kondo effect in the case of a two-level quantum dot coupled to ferromagnetic leads. Using the numerical renormalization-group (NRG) method, we have determined the gate voltage dependence of the dot’s spin and level-resolved spectral functions. We have shown that the polarization of the dot is very susceptible to spin imbalance in the leads and changes sign in the middle of the $S=1$ Coulomb valley. Furthermore, we have also found that by fine tuning an external magnetic field one can compensate for the presence of ferromagnetic leads and restore the Kondo effect in the case of $S=\frac{1}{2}$ Coulomb valley. The underscreened Kondo effect is, however, extremely sensitive to the magnetic field which hinders the full restoration of the Kondo peak.

Figure 1

(Color online) Schematic of a two-level quantum dot coupled to a ferromagnetic reservoir with spin polarization $p$. The dot level energies are denoted by ${\epsilon }_1$ and ${\epsilon }_2$, $\delta$ is the level spacing, while $U^{\prime }$ and $J$ describe the interlevel Coulomb correlations and spin exchange interaction. The $j\text{th}$ dot level is coupled to ferromagnetic electrode with strength ${\Gamma }_{j\sigma }$.

Figure 2

(Color online) (a) The expectation value of the dot spin operator $S^2$ in presence of nonmagnetic and (b) the expectation value of dot spin operator $S^z$ for ferromagnetic lead as a function of the average level position $\epsilon =\left({\epsilon }_1+{\epsilon }_2\right)/2$. The solid line corresponds to $\Gamma =0.04$, while the dashed line corresponds to $\Gamma =0.015$ (in units of $D=1$). The expectation values separately for the two dot levels are also shown. The parameters are $\delta =0.1$, $J=-0.2$, $U=U^{\prime }=0.4$, $B=0$, and in the ferromagnetic case $p=0.4$. The shadowed regions indicate Coulomb valleys with different electron number in the dot, where $N={\sum }_{j\sigma }{n}_{j\sigma }$.

Figure 3

(Color online) [(a)–(c)] The density plots of the normalized level-resolved spectral function $\pi \Gamma A_{j{j}^{\prime }}^{p=0}=\pi {\sum }_{\sigma }{\Gamma }_{\sigma }{A}_{j{j}^{\prime }\sigma }^{p=0}$ and (d) the full spectral function $\pi \Gamma A^{p=0}={\sum }_{j{j}^{\prime }}\pi \Gamma A_{j{j}^{\prime }}^{p=0}$ in the case when the dot is attached to a nonmagnetic electrode $\left(p=0\right)$. The parameters are the same as in Fig. 2 with $\Gamma =0.04$. Note the logarithmic scale on the frequency axis.

Figure 4

(Color online) [(a)–(f)] The density plots of the spin-dependent level-resolved spectral function $\pi {\Gamma }_{\sigma }{A}_{j{j}^{\prime }\sigma }^{p=0.4}$, [(g) and (h)] the spin-dependent full spectral function $\pi {\Gamma }_{\sigma }{A}_{\sigma }^{p=0.4}={\sum }_{j{j}^{\prime }}\pi \Gamma A_{j{j}^{\prime }\sigma }^{p=0.4}$, and (i) the full spectral function $\pi {\sum }_{\sigma }{\Gamma }_{\sigma }{A}_{\sigma }^{p=0.4}$ in the case of ferromagnetic lead with $p=0.4$. The parameters are the same as in Fig. 2 with $\Gamma =0.04$. Note the logarithmic scale on the frequency axis.

Figure 5

(Color online) (a) The spin-up, (b) spin-down, and (c) the full spectral function in the $S=\frac{1}{2}$ Kondo regime for $\epsilon =0$ in the presence of external magnetic field $B$, as indicated in the figure. The parameters are the same as in Fig. 2 with $\Gamma =0.04$ and $p=0.4$. The inset in (c) shows the zoom-out full spectral function. The Kondo effect is restored when $B=0.009215$.

Figure 6

(Color online) (a) The spin-up, (b) spin-down, and (c) the full spectral function in the underscreened Kondo regime for $\epsilon =-0.4$ in the presence of external magnetic field $B$, as indicated in the figure. The parameters are the same as in Fig. 2 with $\Gamma =0.04$ and $p=0.4$. The inset in (c) displays the dependence of the spectral function associated with the Kondo peak.